Chapter 3
Perturbation Theory
In this approximation, the hypothesis on W n is that these are built by known
solutions W
0 of the equation
H
0
W
0
n ¼ E
0
n W
0
n
The hypothesis on H is that
H = H
0 + H
0 where H
0 is the operator associated to the perturbation energy with
respect to pristine H
0 .
The following equation has to be solved
HW n ¼ E n W n
E n ¼ E
0
n þ W
0
n H
0
j jW
0
n
W n ¼ W
0
n þ
X
k6 ¼n
W
0
k H
0
j jW
0
n
 W
0
k =E
0
n À E
0
k
Evidently, the perturbed energy is the addition of the unperturbed one and of the
perturbation energy obtained by the H
0 operator.
The perturbed eigenfunctions are the unperturbed ones modified by the perturbation energy, weighted on the energy differences between the ground state and the
excited interacting states.
Assuming as example a system whose unperturbed eigenfunctions are W
0
1 and
W
0
2 , corresponding to energies E
0
1 and E
0
2
the determinant
H 11 À E H 12
H 21
H 22 À E
© Springer Nature Switzerland AG 2021
F. Morazzoni, Symmetry in Inorganic and Coordination Compounds, Lecture Notes
in Chemistry 106, https://doi.org/10.1007/978-3-030-62996-0_3
39